From the 70es of the 19th century to the first years of the 20th century, Gottlob Frege defined a logic with first and second order concepts (now called "predicates") that would become the archetype of predicate logic as it is known today. In 1901, Betrand Russell defined in Frege's logic a second-order predicate "applying of itself" and expressing a property that cannot exist. This definition is now known as "Russell's Paradox". Russel's and Frege's view was that the existence of such a paradoxical definition ruined Frege's logic. Russell proposed his Theory of Types that, by requiring a predicate of order n to apply only of predicates of orders strictly smaller than n, preclude predicates that "apply of themselves" and therefore prevents paradoxes like Russell's paradox.

This talk first observes that predicates that "apply of themselves" are widespread in computing from relational databases to meta-programming to knowledge representation. An adaptation of the syntax and Herbrand model theory of Ambivalent Logic is introduced, a logic tolerating predicates that "apply of themselves", proposed by Jiang and Kalsbeek in 1994.

Observing that in Ambivalent Logic, paradoxical definitions like that of Russell's paradox are possible, this talk then stresses that this is the notion of a model theory, introduced by Kurt Gödel in 1930 with his doctoral thesis, that makes it possible for us to accept inconsistent definitions like that of Russell's paradox that for Russells and Frege were unacceptable.

Finally, this talk briefly describes a research agenda towards designing and implementing a logic programming language based on Ambivalent Logic.

François Bry, born 1956, works since 1994 as professor with the Institute for Informatics of Ludwig-Maximilian University of Munich, Germany. His current interests include declarative programming, human computation, computing in finances and economics, technology enhanced learning, and ethics of computing. Before joining the University of Munich, he worked from 1985 to 1993 on deductive databases, logic programming, and automated reasoning at ECRC (European Computer-Industry Research Centre), Munich, from 1982 to 1983 on statistic databases at the research center IRT (now INRETS) in Paris, France, and he contributed from 1979 to 1981 to the development of a text processing system at Transac-Alcatel (now Alcatel) in Paris, France. He received a doctoral degree in mathematics in 1981 from the University Paris 6 Pierre et Marie Curie, France.